1. Field of the Invention
This invention relates generally to business forecasting and analysis, and more particularly to determining buy quantities.
2. Description of Related Art
In today's supply chain environment, the newsboy model is the primary tool for dealing with inventory control with stochastic demands. The model is based on the classical one developed originally by Hadley and Whiten (1963), in which one attempts to maximize the profit subject to the constraint on the budget. They offered a Lagrangian method to solve the problem. Since then, various efforts have been made to solve completely the newsboy problem. Yet, no optimal solution has been found with the lower bounds on the ordered quantities.
It is worthy to note that Hadley and Whitin's Lagrangian method relaxes the non-negativity constraints of the order quantities. This would not have serious consequences if the budget is large enough to cover all items. The issue of non-negativity constraint was not paid too much attention until Lau and Lau (1995, 1996) who observed that relaxing the non-negativity constraints could lead to negative order quantities for some of the considered products. On the other hand, if the non-negativity constraints are not relaxed and Kuhn-Tucker conditions are applied, the number of non-linear equations to be solved simultaneously grows exponentially as the number of products increases. This could be one of the reasons that most existing models relax the lower bounds to make the problem tractable.
An important advance to the newsboy problem is discussed in Abdel-Malek, et al., “An analysis of the multi-product newsboy problem with a budget constraint,” International Journal of Production Economics, Volume 97, Issue 3, 18 Sep. 2005, Pages 296-307, which is hereby incorporated by reference, and in Abdel-Malek, et al., “On the multi-product newsboy problem with two constraints,” Computers & Operations Research, Volume 32, Issue 8, August 2005, Pages 2095-2116, which is hereby incorporated by reference. This approach involves dividing the available budget into three regions as illustrated in FIG. 1. The first region corresponds to the case where the budget is large enough to order the optimum quantity of each item. The second region corresponds to the case where a budget constraint is binding. The third region corresponds to the case where the budget is not large enough to order all of the products. A distinct solution is then determined for each region of the available budget. For the first region, the solution yields the global maxima of the expected profit and the non-negativity constraints are not binding. For the second region, the Lagrangian approach with relaxed lower bounds can be used to determine a solution. Depending on the type of demand distribution for each product, one can choose from exact, approximate, or Generic Iterative Method solution models to obtain the lot size for each product. For the third region, the approach is based on a duality theory and starts by deleting, in ascending order, products with lower marginal utilities at their bounds until the remaining products can fit within the available budget. Then, the non-negativity constraints are relaxed and one can apply one of the solution methods followed in the second region.
Deleting products with lower marginal utilities at their bounds cannot guarantee that the least efficient products will be eliminated. For example, low marginal utility at ordered quantity Qi=0 does not necessarily mean low marginal utility at other points Qi>0. Thus, while this method is sophisticated and intuitively appealing, it does not offer an optimal solution to the newsboy problem.